Pub Date : 2023-05-08DOI: 10.48550/arXiv.2305.04563
D. Musatov, Georgii Potapov
This is the full version of a paper submitted to the Computability in Europe (CiE 2023) conference, with all proofs omitted there. In 2012 P. D. Azar and S. Micali introduced a new model of interactive proofs, called"Rational Interactive Proofs". In this model the prover is neither honest nor malicious, but rational in terms of maximizing his expected reward. In this article we explore the connection of this area with classic complexity results. In the first part of this article we revise the ties between the counting hierarchy and the hierarchy of constant-round rational proofs. We prove that a polynomial-time machine with oracle access to DRMA[k] decides exactly languages in DRMA[k], a coincidence unknown for levels of the counting hierarchy. In the second part we study communication complexity of single-round rational proofs. We show that the class defined by logarithmic-communication single-round rational proofs coincides with PP. We also show that single-round rational protocols that treat problems in Parity-P as black-box samplers of a random variable require at least a linear number of bits of communication.
这是提交给欧洲可计算性(CiE 2023)会议的一篇论文的完整版本，其中省略了所有证明。2012年，P. D. Azar和S. Micali引入了一种新的交互证明模型，称为“理性交互证明”。在这个模型中，证明者既不诚实也不恶意，但在最大化他的预期回报方面是理性的。在本文中，我们将探讨这一领域与经典复杂性结果的联系。在本文的第一部分中，我们修正了计数层次与常轮有理证明层次之间的联系。我们证明了对DRMA[k]具有oracle访问权限的多项式时间机器准确地决定了DRMA[k]中的语言，这是计数层次结构级别未知的巧合。第二部分研究了单轮有理证明的通信复杂性。我们表明，由对数通信单轮有理证明定义的类与PP一致。我们还表明，将奇偶性- p中的问题视为随机变量的黑盒采样的单轮有理协议至少需要线性数量的通信位。
{"title":"Structural Complexity of Rational Interactive Proofs","authors":"D. Musatov, Georgii Potapov","doi":"10.48550/arXiv.2305.04563","DOIUrl":"https://doi.org/10.48550/arXiv.2305.04563","url":null,"abstract":"This is the full version of a paper submitted to the Computability in Europe (CiE 2023) conference, with all proofs omitted there. In 2012 P. D. Azar and S. Micali introduced a new model of interactive proofs, called\"Rational Interactive Proofs\". In this model the prover is neither honest nor malicious, but rational in terms of maximizing his expected reward. In this article we explore the connection of this area with classic complexity results. In the first part of this article we revise the ties between the counting hierarchy and the hierarchy of constant-round rational proofs. We prove that a polynomial-time machine with oracle access to DRMA[k] decides exactly languages in DRMA[k], a coincidence unknown for levels of the counting hierarchy. In the second part we study communication complexity of single-round rational proofs. We show that the class defined by logarithmic-communication single-round rational proofs coincides with PP. We also show that single-round rational protocols that treat problems in Parity-P as black-box samplers of a random variable require at least a linear number of bits of communication.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130825370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-07DOI: 10.48550/arXiv.2305.04234
A. Barsukov, Florent R. Madelaine
We investigate logics and classes of problems below Fagin's existential second-order logic (ESO) and above Feder and Vardi's logic for constraint satisfaction problems (CSP), the so called monotone monadic SNP without inequality (MMSNP). It is known that MMSNP has a dichotomy between P and NP-complete but that the removal of any of these three restrictions imposed on SNP yields a logic that is Ptime equivalent to ESO: so by Ladner's theorem we have three stronger sibling logics that are nondichotomic above MMSNP. In this paper, we explore the area between these four logics, mostly by considering guarded extensions of MMSNP, with the ultimate goal being to obtain logics above MMSNP that exhibit such a dichotomy.
{"title":"On guarded extensions of MMSNP","authors":"A. Barsukov, Florent R. Madelaine","doi":"10.48550/arXiv.2305.04234","DOIUrl":"https://doi.org/10.48550/arXiv.2305.04234","url":null,"abstract":"We investigate logics and classes of problems below Fagin's existential second-order logic (ESO) and above Feder and Vardi's logic for constraint satisfaction problems (CSP), the so called monotone monadic SNP without inequality (MMSNP). It is known that MMSNP has a dichotomy between P and NP-complete but that the removal of any of these three restrictions imposed on SNP yields a logic that is Ptime equivalent to ESO: so by Ladner's theorem we have three stronger sibling logics that are nondichotomic above MMSNP. In this paper, we explore the area between these four logics, mostly by considering guarded extensions of MMSNP, with the ultimate goal being to obtain logics above MMSNP that exhibit such a dichotomy.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126622030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-29DOI: 10.48550/arXiv.2305.00308
P. Parys, Aleksander Wiacek
We improve the complexity of solving parity games (with priorities in vertices) for $d={omega}(log n)$ by a factor of ${theta}(d^2)$: the best complexity known to date was $O(mdn^{1.45+log_2(d/log_2(n))})$, while we obtain $O(mn^{1.45+log_2(d/log_2(n))}/d)$, where $n$ is the number of vertices, $m$ is the number of edges, and $d$ is the number of priorities. We base our work on existing algorithms using universal trees, and we improve their complexity. We present two independent improvements. First, an improvement by a factor of ${theta}(d)$ comes from a more careful analysis of the width of universal trees. Second, we perform (or rather recall) a finer analysis of requirements for a universal tree: while for solving games with priorities on edges one needs an $n$-universal tree, in the case of games with priorities in vertices it is enough to use an $n/2$-universal tree. This way, we allow to solve games of size $2n$ in the time needed previously to solve games of size $n$; such a change divides the quasi-polynomial complexity again by a factor of ${theta}(d)$.
{"title":"Improved Complexity Analysis of Quasi-Polynomial Algorithms Solving Parity Games","authors":"P. Parys, Aleksander Wiacek","doi":"10.48550/arXiv.2305.00308","DOIUrl":"https://doi.org/10.48550/arXiv.2305.00308","url":null,"abstract":"We improve the complexity of solving parity games (with priorities in vertices) for $d={omega}(log n)$ by a factor of ${theta}(d^2)$: the best complexity known to date was $O(mdn^{1.45+log_2(d/log_2(n))})$, while we obtain $O(mn^{1.45+log_2(d/log_2(n))}/d)$, where $n$ is the number of vertices, $m$ is the number of edges, and $d$ is the number of priorities. We base our work on existing algorithms using universal trees, and we improve their complexity. We present two independent improvements. First, an improvement by a factor of ${theta}(d)$ comes from a more careful analysis of the width of universal trees. Second, we perform (or rather recall) a finer analysis of requirements for a universal tree: while for solving games with priorities on edges one needs an $n$-universal tree, in the case of games with priorities in vertices it is enough to use an $n/2$-universal tree. This way, we allow to solve games of size $2n$ in the time needed previously to solve games of size $n$; such a change divides the quasi-polynomial complexity again by a factor of ${theta}(d)$.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"18 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125908325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-26DOI: 10.48550/arXiv.2304.13550
Pacôme Perrotin, Sylvain Sen'e
We state an algorithm that, given an automata network and a block-sequential update schedule, produces an automata network of the same size or smaller with the same limit dynamics under the parallel update schedule. Then, we focus on the family of automata cycles which share a unique path of automata, called tangential cycles, and show that a restriction of our algorithm allows to reduce any instance of these networks under a block-sequential update schedule into a smaller parallel network of the family and to characterize the number of reductions operated while conserving their limit dynamics. We also show that any tangential cycles reduced by our main algorithm are transformed into a network whose size is that of the largest cycle of the initial network. We end by showing that the restricted algorithm allows the direct characterization of block-sequential double cycles as parallel ones.
{"title":"Turning block-sequential automata networks into smaller parallel networks with isomorphic limit dynamics","authors":"Pacôme Perrotin, Sylvain Sen'e","doi":"10.48550/arXiv.2304.13550","DOIUrl":"https://doi.org/10.48550/arXiv.2304.13550","url":null,"abstract":"We state an algorithm that, given an automata network and a block-sequential update schedule, produces an automata network of the same size or smaller with the same limit dynamics under the parallel update schedule. Then, we focus on the family of automata cycles which share a unique path of automata, called tangential cycles, and show that a restriction of our algorithm allows to reduce any instance of these networks under a block-sequential update schedule into a smaller parallel network of the family and to characterize the number of reductions operated while conserving their limit dynamics. We also show that any tangential cycles reduced by our main algorithm are transformed into a network whose size is that of the largest cycle of the initial network. We end by showing that the restricted algorithm allows the direct characterization of block-sequential double cycles as parallel ones.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114061037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-14DOI: 10.48550/arXiv.2302.07249
P. Arrighi, Amélia Durbec, Pierre Guillon
We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the question whether such local conditions can enforce a specific support graph, and thus relate the model to classical symbolic dynamics. We prove that the subshifts that contain only infinite graphs are either aperiodic, or feature no residual finiteness of their period group, yielding non-trivial examples as well as two natural undecidability theorems.
{"title":"Graph subshifts","authors":"P. Arrighi, Amélia Durbec, Pierre Guillon","doi":"10.48550/arXiv.2302.07249","DOIUrl":"https://doi.org/10.48550/arXiv.2302.07249","url":null,"abstract":"We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the question whether such local conditions can enforce a specific support graph, and thus relate the model to classical symbolic dynamics. We prove that the subshifts that contain only infinite graphs are either aperiodic, or feature no residual finiteness of their period group, yielding non-trivial examples as well as two natural undecidability theorems.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132588956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-14DOI: 10.48550/arXiv.2302.07066
Sam Sanders
Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $exists^{n}$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied, based on well-known theorems like the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $exists^{2}$ while the former are only computable in $exists^{3}$. While there is a great divide separating $exists^{2}$ and $exists^{3}$, we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.
{"title":"The non-normal abyss in Kleene's computability theory","authors":"Sam Sanders","doi":"10.48550/arXiv.2302.07066","DOIUrl":"https://doi.org/10.48550/arXiv.2302.07066","url":null,"abstract":"Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $exists^{n}$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied, based on well-known theorems like the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $exists^{2}$ while the former are only computable in $exists^{3}$. While there is a great divide separating $exists^{2}$ and $exists^{3}$, we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132993835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-13DOI: 10.1007/978-3-031-36978-0_6
L. Galeotti, Ethan Lewis, B. Löwe
{"title":"Symmetry for Transfinite Computability","authors":"L. Galeotti, Ethan Lewis, B. Löwe","doi":"10.1007/978-3-031-36978-0_6","DOIUrl":"https://doi.org/10.1007/978-3-031-36978-0_6","url":null,"abstract":"","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116889800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-15DOI: 10.48550/arXiv.2209.07243
A. Shen
We show that linear inequalities for entropies have a natural geometric interpretation in terms of Hausdorff and packing dimensions, using the point-to-set principle and known results about inequalities for complexities, entropies and the sizes of subgroups.
{"title":"Inequalities for entropies and dimensions","authors":"A. Shen","doi":"10.48550/arXiv.2209.07243","DOIUrl":"https://doi.org/10.48550/arXiv.2209.07243","url":null,"abstract":"We show that linear inequalities for entropies have a natural geometric interpretation in terms of Hausdorff and packing dimensions, using the point-to-set principle and known results about inequalities for complexities, entropies and the sizes of subgroups.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131801083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-19DOI: 10.48550/arXiv.2208.09373
Zeev Nutov
In minimum power network design problems we are given an undirected graph $G=(V,E)$ with edge costs ${c_e:e in E}$. The goal is to find an edge set $Fsubseteq E$ that satisfies a prescribed property of minimum power $p_c(F)=sum_{v in V} max {c_e: e in F mbox{ is incident to } v}$. In the Min-Power $k$ Edge Disjoint $st$-Paths problem $F$ should contains $k$ edge disjoint $st$-paths. The problem admits a $k$-approximation algorithm, and it was an open question whether it admits approximation ratio sublinear in $k$ even for unit costs. We give a $4sqrt{2k}$-approximation algorithm for general costs.
在最小电网设计问题中，我们给出了一个无向图$G=(V,E)$，其边缘代价为${c_e:e in E}$。目标是找到一个边集$Fsubseteq E$，它满足最小幂$p_c(F)=sum_{v in V} max {c_e: e in F mbox{ is incident to } v}$的规定性质。在最小功率$k$边不相交$st$ -路径问题中$F$应该包含$k$边不相交$st$ -路径。这个问题承认$k$近似算法，它是否承认近似比在$k$的次线性是一个开放的问题，即使是单位成本。我们给出了一般成本的$4sqrt{2k}$ -近似算法。
{"title":"An $O(sqrt{k})$-approximation algorithm for minimum power $k$ edge disjoint $st$ -paths","authors":"Zeev Nutov","doi":"10.48550/arXiv.2208.09373","DOIUrl":"https://doi.org/10.48550/arXiv.2208.09373","url":null,"abstract":"In minimum power network design problems we are given an undirected graph $G=(V,E)$ with edge costs ${c_e:e in E}$. The goal is to find an edge set $Fsubseteq E$ that satisfies a prescribed property of minimum power $p_c(F)=sum_{v in V} max {c_e: e in F mbox{ is incident to } v}$. In the Min-Power $k$ Edge Disjoint $st$-Paths problem $F$ should contains $k$ edge disjoint $st$-paths. The problem admits a $k$-approximation algorithm, and it was an open question whether it admits approximation ratio sublinear in $k$ even for unit costs. We give a $4sqrt{2k}$-approximation algorithm for general costs.","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133292155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-08DOI: 10.1007/978-3-031-08740-0_3
I. Blechschmidt, P. Schuster
{"title":"Maximal Ideals in Countable Rings, Constructively","authors":"I. Blechschmidt, P. Schuster","doi":"10.1007/978-3-031-08740-0_3","DOIUrl":"https://doi.org/10.1007/978-3-031-08740-0_3","url":null,"abstract":"","PeriodicalId":436783,"journal":{"name":"Conference on Computability in Europe","volume":"55 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131445741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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